TCP Takes care of client identification
As a.m. said, TCP takes care of the client identification, and the server only sees a "socket" per client.
Say a server at 10.10.100.100 listens to port 80 for incoming TCP connections (HTTP is built over TCP). A client's browser (at 10.9.8.7) connects to the server using the client port 27143. The server sees: "the client 10.9.8.7:27143 wants to connect, do you accept?". The server app accepts, and is given a "handle" (a socket) to manage all communication with this client, and the handle will always send packets to 10.9.8.7:27143 with the proper TCP headers.
Packets are never simultaneous
Now, physically, there is generally only one (or two) connections linking the server to internet, so packets can only arrive in sequential order. The question becomes: what is the maximum throughput through the fiber, and how many responses can the server compute and send in return. Other than CPU time spent or memory bottlenecks while responding to requests, the server also has to keep some resources alive (at least 1 active socket per client) until the communication is over, and therefore consume RAM. Throughput is achieved via some optimizations (not mutually-exclusive): non-blocking sockets (to avoid pipelining/socket latencies), multi-threading (to use more CPU cores/threads).
Improving request throughput further: load balancing
And last, the server on the "front-side" of websites generally do not do all the work by themselves (especially the more complicated stuff, like database querying, calculations etc.), and defer tasks or even forward HTTP requests to distributed servers, while they keep on handling trivially (e.g. forwarding) as many requests per second as they can. Distribution of work over several servers is called load-balancing.